Differential Geometry of Singular Spaces

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The study of Riemannian manifolds is central in mathematics. They are smooth objects which can be analyzed using geometric and analytic tools. It turned out over the last fifteen years that the study of certain limits of Riemannian manifolds is very fruitful. In general, these limits present difficult singularities. Examples for limit spaces are metric spaces with a lower or an upper curvature bound. In this project, we want to use Geometric Measure Theory in order to construct and study other types of limit spaces which are better suited for questions about scalar curvature and Ricci curvature. There will be an interesting interplay between Differential Geometry, Geometric Measure Theory, Convex Geometry and Subanalytic Geometry. Hopefully, there will be applications of this theory to the existence of Einstein metrics, realization of Yamable invariants and to the geometry of Riemannian manifolds with scalar or Ricci curvature bounds.